Question

decay rate
half - life
15.6 days
complete the table shown to the right for the half - life of a certain radioactive substance.
k =
(round to six decimal places as needed.)

Explanation:

Step1: Recall decay formula

The formula for exponential decay is \( N(t) = N_0 e^{-kt} \), and for half - life (\( t = T_{1/2} \)), \( N(T_{1/2})=\frac{N_0}{2} \). Substituting into the decay formula: \( \frac{N_0}{2}=N_0 e^{-kT_{1/2}} \).

Step2: Solve for k

Divide both sides by \( N_0 \): \( \frac{1}{2}=e^{-kT_{1/2}} \). Take the natural logarithm of both sides: \( \ln(\frac{1}{2})=-kT_{1/2} \). Then, \( k =-\frac{\ln(\frac{1}{2})}{T_{1/2}}=\frac{\ln(2)}{T_{1/2}} \). Given \( T_{1/2} = 15.6 \) days, substitute \( T_{1/2} \) into the formula: \( k=\frac{\ln(2)}{15.6} \). Calculate \( \ln(2)\approx0.693147 \), so \( k=\frac{0.693147}{15.6}\approx0.044432 \).

Answer:

\( k\approx0.044432 \)